# Multivariable Calculus/Calculus & Geometry textbook suggestions

Hello all,

I am currently in University and I am retaking a course that I dropped last year due to the fact that there was no textbook and the time I spent in the course was a frantic scramble for reliable information and attempting to understand what the professor was teaching. I have looked before and the only textbook that I found that was remotely close to the material that is covered in the course is Multivariable Calculus by Edwards and Penney. I am asking the community for any suggestions for textbooks that cover most, if not all of the topics listed below:

- Differentiation in Several Variables
Partial derivatives, directional derivatives, del operator, Mean-value theorem for a function of several variables, differentiation through an integral, Leibniz's Rule 2nd order derivatives and Clairaut's theorem, Hessian Matrix. Functions from several variables to several variables, Jacobian Matrix

- Inverse - and Implicit functions in several variables
Inverse function theorem
Implicit function theorem

Matrix Representation
Change of variable & diagonalization by congruence
Positive & Negative definiteness, Semi-definieness, determinantal criteria
Sylvester's Theorem

- Extrema
Local extrema-critical points, local max, local min, saddle points, determinantal tests on the Hessian Matrix
Global extrema-existence for continuous functwions on closed, bounded sets, finiding constrained extrema by the method of Lagrange multipliers

- Curves and Surfaces
Space curves defined parametrically-tangent, normal, binormal, arc length, curvature, torsion
Surfaces-implicit and parametric definitions-tangent plane, normal line
Space curves defined as intersecting surgaces, related quadratic & cubic forms

- Integration in several Variables
Multiple and iterated integrals
Change of order of variables
Change of variable formula in general
Polar, cylindrical & spherical coordinates
Line integrals, consecutive and non-consecutive vector fields, curl operator
Surface integrals, projection onto a plane, parametric surgace integrals
Stokes' theorem, boundary curve, orientation
Gauss' theorem, boundary surgace, div operator

Any suggestions will be greatly appreciated and sorry for the long list, I didn't want to leave anything out.

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I fully second Apostol and Courant.

Apostol and Courant are excellent but they are mathematically rigorous and I don't know if that is what you want. If not, then there is the book by Schey Div, Grad, Curl and All That for intuition; Marsden and Tromba Vector Calculus is an intermediate sort of book; and the old favourite is Stewart Multivariable Calculus or the relevant chapters in his Calculus. Stewart gives the simplest presentation of calculus.

I suggest going to the library and looking through all these books and deciding for yourself which one best suites your needs.

any multivariable calculus text will fit your description..

I agree that Courant and Apostol are NOT light reads. Stewart has an excellent MV book that is good for all audiences. A lesser known book is Howard Anton's Calculus, A new horizon, which has a lot of good applications of calc added in. Of course the Khan Academy has very easy-to-understand lectures, but not in all of the above mentioned topics.